How structure sculpts function: Unveiling the contribution of anatomical connectivityto the brain's spontaneous correlation structureR. G. Bettinardi, G. Deco, V. M. Karlaftis, T. J. Van Hartevelt, H. M. Fernandes, Z. Kourtzi, M. L. Kringelbach, andG. Zamora-López

Citation: Chaos 27, 047409 (2017); doi: 10.1063/1.4980099View online: http://dx.doi.org/10.1063/1.4980099View Table of Contents: http://aip.scitation.org/toc/cha/27/4Published by the American Institute of Physics

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How structure sculpts function: Unveiling the contribution of anatomicalconnectivity to the brain’s spontaneous correlation structure

R. G. Bettinardi,1,2,a) G. Deco,1,2,3 V. M. Karlaftis,4 T. J. Van Hartevelt,5,6

H. M. Fernandes,5,6 Z. Kourtzi,4 M. L. Kringelbach,5,6 and G. Zamora-L�opez1,2,b)

1Center for Brain and Cognition, Universitat Pompeu Fabra, Barcelona, Spain2Department of Information and Communication Technologies, Universitat Pompeu Fabra, Barcelona, Spain3Instituci�o Catalana de la Recerca i Estudis Avancats, Universitat Pompeu Fabra, Barcelona, Spain4Department of Psychology, University of Cambridge, Cambridge, United Kingdom5Department of Psychiatry, University of Oxford, Oxford, United Kingdom6Center for Music in the Brain, Aarhus University, Aarhus, 8000 Aarhus C, Denmark

(Received 1 November 2016; accepted 3 April 2017; published online 17 April 2017)

Intrinsic brain activity is characterized by highly organized co-activations between different

regions, forming clustered spatial patterns referred to as resting-state networks. The observed co-

activation patterns are sustained by the intricate fabric of millions of interconnected neurons consti-

tuting the brain’s wiring diagram. However, as for other real networks, the relationship between the

connectional structure and the emergent collective dynamics still evades complete understanding.

Here, we show that it is possible to estimate the expected pair-wise correlations that a network

tends to generate thanks to the underlying path structure. We start from the assumption that in order

for two nodes to exhibit correlated activity, they must be exposed to similar input patterns from the

entire network. We then acknowledge that information rarely spreads only along a unique route but

rather travels along all possible paths. In real networks, the strength of local perturbations tends to

decay as they propagate away from the sources, leading to a progressive attenuation of the original

information content and, thus, of their influence. Accordingly, we define a novel graph measure,

topological similarity, which quantifies the propensity of two nodes to dynamically correlate as a

function of the resemblance of the overall influences they are expected to receive due to the under-

lying structure of the network. Applied to the human brain, we find that the similarity of whole-

network inputs, estimated from the topology of the anatomical connectome, plays an important role

in sculpting the backbone pattern of time-average correlations observed at rest. VC 2017 Author(s).All article content, except where otherwise noted, is licensed under a Creative CommonsAttribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).[http://dx.doi.org/10.1063/1.4980099]

The quest to understand how structure shapes function

lies at the heart of a broad spectrum of disciplines, ranging

from biology to network science. For over a decade,

many efforts have been devoted to investigating the

impact of different network features, e.g., hubs, clustering,

or communities, on the collective behaviour of dynamical

processes on complex networks such as spreading phe-

nomena and synchronization. However, a unique answer

to this question is not possible, because the emerging net-

work activity is a product of the interplay between the net-

work’s topology, the particular local dynamics governing

nodes’ behavior, and the coupling function defining how

information is transferred: network’s topology shapes, but

does not determine, the collective dynamics. The question

is thus whether we can estimate what is the contribution of

the structure alone, and which are the most relevant topo-

logical features in sculpting the emergent functional rela-

tions. Here, we have shown that the global path structure

of the network is what truly determines the contribution

of the network over the collective dynamics, as it implicitly

incorporates information about all other network features,

e.g., degree-distributions or modules. The expected magni-

tude of synchrony or correlation between two nodes is

largely governed by the common inputs they receive from

all other nodes, given that information propagates along

all possible paths of any length. We quantify this pair-

wise, whole-network affinity introducing a network mea-

sure, the topological similarity (T ). Formally, T is the

direct relation between the structure of a network and the

pattern of functional relations that it tends to produce.

Applied to the human brain, we find that the similarity of

whole-network inputs, defined by the topology of the

underlying anatomical connectome, plays an important

role in sculpting the backbone pattern of time-average

correlations observed at rest. This confirms the pivotal rel-

evance of the path structure in sculpting the network’s

correlations due to spontaneous activity.

I. INTRODUCTION

The quest to understand how structure shapes function

lies at the heart of a broad spectrum of disciplines, ranging

from molecular biology to network science. A classic example

a)Electronic mail: ruggero.bettinardi@upf.edub)Electronic mail: gorka@Zamora-Lopez.xyz

1054-1500/2017/27(4)/047409/12 VC Author(s) 2017.27, 047409-1

CHAOS 27, 047409 (2017)

in biology is to predict the functional relevance of proteins’

three-dimensional structure,1,2 whereas in network science,

the challenge is to isolate those graph features which drive the

interaction patterns arising when a complex network hosts a

dynamical process, such as synchronization3,4 and spreading

phenomena.5,6 However, it might not be possible to explain

emergent network behavior just as a function of its underlying

architecture, as this results from the inextricable interplay

between structural and dynamical factors, e.g., those control-

ling intrinsic properties of the nodes.7–10

The relationship between the structure and the function

is particularly relevant in neuroscience, as it has been repeat-

edly shown that morphological and structural variations of

the nervous system tend to be associated with behavioral

changes due to alterations of intrinsic brain activity’s organi-

zation.11–13 In the last two decades, a large body of research

has demonstrated that spontaneous brain fluctuations form

structured patterns of consistent co-activations across differ-

ent subsets of regions.14–18 Many efforts have been devoted

to reproducing resting-state brain activity by means of com-

putational modeling. Early models based on the structural

connectomes of cats and macaques explored the emerging

patterns of correlations in those networks at different spatial

and temporal scales.19–21 With the arrival of structural

human connectomes obtained via diffusion imaging and trac-

tography, computational models could be validated with the

empirical correlation structure observed in the human

resting-state, referred to as functional connectivity (FC).22–26

Systematic comparisons showed that using different models

to simulate the activity of brain regions returns correlation

matrices of varying accuracy to fit the empirical ones.10,27

So far, we still lack a unitary model of the relationship

between the shape of the brain’s connectome and the emer-

gent activity patterns. One of the main reasons is that interac-

tions between different areas do not only depend on the

structure of the connectome but also on the local and global

dynamics characterizing a given brain state, such as rest,

sleep, and anesthesia.28–30 The existence of different state-

dependent activity patterns sustained by the same underlying

anatomy exposes how elusive is the relationship between the

network structure and function.

In the present paper, we aim at unveiling what is the

expected contribution of network topology on the pattern of

interactions it naturally tends to generate and sustain. To this

aim, in Section II, we summarize the main topological fea-

tures that contribute to the routing of information through

the network and introduce a novel graph theoretical quantity,

T , measuring the similarity of the entire input profiles that

two nodes receive from the whole network. This measure,

that we named topological similarity, is a generalization of

the concept of matching index that explicitly accounts for

the fact that, in networks, information travels along all possi-

ble paths, not only along the shortest ones, and that its con-

tent tends to decay as it moves away from the source.31–34

This measure, based on a network’s topology, represents the

expected time-average correlation structure of the network

due to topological constraints.

In Section III, we systematically investigate the contri-

bution of three fundamental topological features, namely, the

weight of the links, the length of the path, and the presence

of redundant alternative paths. Together, they specify the

graph’s path structure, which determines how influence

spreads through the network and, as a consequence, sculpts

the inputs that nodes receive.

Finally, in Section IV, we investigate the contribution of

the anatomical connectivity of the human brain to its sponta-

neous correlation structure. For this, we have calculated

our topological similarity measure out of the empirically

obtained structural connectome, and we have considered

it as a zeroth-order approximation of the cross-correlation

matrix that one could expect due to topological constraints.

We find that the topological similarity captures a consider-

able portion of the empirical functional connectivity mea-

sured with resting-state fMRI. Then, we ran simulations of

the brain’s activity by modeling the local region dynamics

using the Hopf normal model. The numerically simulated

functional connectivity resembles the empirical one more

accurately than the topological similarity does. These results

corroborate that the anatomical connectivity shapes but does

not fully determine the empirical functional connectivity, as

the underlying topology, defined by the complex fabric of

brain axonal pathways, is only one of the factors leading to

the observed collective dynamics.

II. HOW TOPOLOGY SCULPTS NETWORKINTERACTIONS

When addressing the relationship between the structure

and the function in complex networks, we need to remember

that the collective behavior of a set of coupled dynamical

units depends on three principal ingredients: (i) the structure

of the network, (ii) the local dynamics of the nodes, and (iii)the coupling function determining how information is passed

from one node to another. In fact, for a fixed network, chang-

ing the local dynamical model of the nodes and the coupling

function usually leads to different collective dynamics.8–10

Therefore, in order to estimate the contribution of the struc-

ture alone we need to isolate, as much as possible, its contri-

bution from that of the other two factors.

Typically, the activity of two nodes exhibits a statistical

dependence either if they are connected by means of a direct

link or if the aggregate input they receive from the entire net-

work is similar, independent of whether there is a link

between them or not. Because information in a network

rarely travels exclusively along the shortest paths35,36 but

instead diffuses along the whole network, the total influence

of one node over another due to topological constraints

mainly depends on three features: (a) the strength of the cou-

pling between them, usually represented by the weights of

the links, (b) the graph distance between the two nodes, and

(c) the presence of multiple, alternative, and re-entrant paths

between the nodes through which information can

travel.37–39 We refer to these three features as the topologicalprimitives because, in combination, they characterize the

path structure of the network.

In general, the influence of a direct link is greater than

the influence exerted over longer indirect paths. In real sys-

tems, the “power” of the signals or their information content

047409-2 Bettinardi et al. Chaos 27, 047409 (2017)

naturally decays along the path,31–34 unless there exists an

active mechanism which amplifies the signal at the cost of

energy. In addition, it is unlikely that the influence or infor-

mation propagates only along a single, selected path, unless

specific gating mechanisms exist to control the routing of

information among all existing paths. The total number

of paths (non-Hamiltonian walks) of length l between two

nodes grows with l. This number is given exactly by the lthpower of the adjacency matrix A, see Ref. 40. The total num-

ber of paths leaving from node i and arriving at node j is

given by the sum

X1

l¼0

ðAlÞij ¼ 1þ Aij þ ðA2Þij þ ðA3Þij þ � � � : (1)

This number typically diverges and thus, for the dynamics

within a network to remain bounded, the amount of influ-ence needs to decay faster with the length than the growth

in the number of paths. Mathematically, the problem con-

sists in finding a set of coefficients fblg for which the seriesP1l¼0 bl Al converges. A solution to this problem is the com-

municability C proposed by Estrada and Hatano.41 This

measure corresponds to the matrix exponential of A, which

can be expanded into a series of powers with coefficients

bl ¼ 1= l!

C � eA ¼X1

l¼0

Al

l!¼ 1þ Aþ A2

2!þ A3

3!þ � � � : (2)

From a physical perspective, the communicability is analo-

gous to the Green’s function of the network41,42 and

expresses how local perturbations propagate along the sys-

tem. Communicability can be tuned using a constant global

parameter g, that uniformly scales the weights of all links in

A, allowing to search over multiple scales42,43

C ¼ egA ¼X1

l¼0

glAl

l!¼ 1þ gAþ g2A2

2!þ g3A3

3!þ � � � (3)

When g is weak, perturbations quickly decay, producing

local correlations only around the node’s neighborhood.

As g grows, perturbations propagate deeper into the

network, giving rise to stronger correlations over more

distant nodes.

The goal of the present work was to address whether it

is possible to use information of the topological properties of

a network to estimate the most likely correlation structure it

tends to exhibit. As argued before, the statistical dependence

of two nodes relies not only on the presence of a direct link

between them but also on the similarity of the common

inputs they are exposed to. We noted that each column vec-

tor cj of the communicability matrix C represents the input

profile of the influences a node receives from all other nodes

along all possible paths. This includes the influence a node

exerts on itself through recurrent (or re-entrant) paths.

Therefore, if the network hosts a dynamical process, the sim-

ilarity of the input profiles ci and cj of nodes i and j could be

regarded as an estimate of their expected interdependence.

Consequently, we define the topological similarity, T ij, as

the cosine similarity between the input profiles of a pair of

nodes

T ij ¼hci; cjikcikkcjk

; (4)

where h ; i is the scalar product and k � k the vector norm.

The definition of T depends uniquely on the topological con-

straints of the network encoded in the adjacency matrix A,

together with the realistic assumption (embedded in C) that

the influence or the information content decays with the

length of the path. Additionally, we find that T is equivalent

to the correlation matrix R of a coupled Gaussian noise

diffusion process in which the matrix exponential is consid-

ered as the kernel of noise propagation,43 see supplementary

material.

The topological similarity can be estimated for both

directed and undirected graphs, as well as for weighted

networks. In the case of undirected graphs, C is symmetric,

but if the links are directed, then C is asymmetric and its

columns determine the input profiles of the nodes while its

rows represent the profile of output influences. Despite

the measure can be computed for any weighted adjacency

matrix, it does not always make sense to do so. Because Tis based on measuring the influence spreading along differ-

ent paths, it only has a physical meaning when the weights

of the links quantify their potential flow capacity or cou-

pling strength. On the other hand, it lacks a physical inter-

pretation when link weights represent other quantities or

statistical associations. For example, while it is common in

the literature to apply graph measures to binarized func-

tional connectivity matrices, T shall not be calculated in

such cases.

In Section III, we systematically investigate how three

network features (the weights of the links, the path length,

and the redundancy of paths) mould both the communicabil-

ity and, as a consequence, the topological similarity between

nodes.

III. ALTERING THE PATH STRUCTURE

We now focus on investigating how different topologi-

cal features of the network modulate the influence that a

node exerts over another. To this aim, we will focus on three

simple classes of graphs: chains, cycles, and path-redundant

motifs, see examples in Fig. 1. We will explore how manipu-

lating crucial parameters of these simple graphs leads to

changes in the influence between selected pairs of nodes

(measured by their communicability, Cij) and in their topo-

logical similarity T ij. In the case of single link motifs, the

contribution of the link’s weight is trivial: the weight modu-

lates the mutual influence between the two nodes (see sup-

plementary material Figure S1). The role of the path length

is best understood studying simple chain topologies of vary-

ing sizes. Increasing the length of the path separating two

nodes leads to a decrease in both their communicability and

topological similarity, Fig. 1(a). This behavior is directly

determined by the decay embedded in the definition of the

communicability. From the example, it is indeed evident

047409-3 Bettinardi et al. Chaos 27, 047409 (2017)

that, for increasing lengths of the chain, the input profiles

of the two end nodes become more and more antithetic, due

to opposed whole-network influences (see the input profiles

highlighted by green rectangles on the communicability

matrices, Fig. 1(a)). As a consequence, their topological sim-

ilarity decreases with path length as well. A uniform varia-

tion of the weights of all links in the chain, which

corresponds to multiply the links by a constant factor, has

mainly a quantitative effect on the decay: increasing the

links’ weights enhances how far and strong the influence of

nodes can travel, Fig. 1(b). The interaction between the chain

length and links’ weight is shown in supplementary material

Figure S2.

Chains can be thought as a baseline to compare more

complex motifs. In fact, cycles and path-redundant topolo-

gies are built upon chain motifs. We now compare Cij

and T ij of the three model graphs (chains, cycles, and

path-redundant motifs) having an identical diameter, i.e.,

length of the longest path. Figure 1(c) provides a schematic

representation of different motifs of the same longest paths.

For the case of chains and path-redundant topologies, we

computed Cij and T ij for the two nodes at the extremes. For

cyclic topologies, we selected two adjacent nodes. This

choice allowed us to disentangle the contribution of the indi-

rect paths above and beyond the modulation produced by the

direct links.

FIG. 1. Behavior of Communicability and Topological Similarity in simple network motifs. (a) Communicability and Topological Similarity matrices of

chains of different lengths. In the upper matrices, the red dots indicate the matrix entry corresponding to the communicability between the nodes at the

two ends of the chains, whereas the green rectangles mark their whole-network input profiles (column vectors), which are used to calculate the topological

similarity of the corresponding nodes (marked with the green dots in the lower matrices. (b) Communicability and Topological Similarity matrices of

chains of constant length (L ¼ 21) for different links’ weights, w. See (a) for the legend of red dots, green rectangles, and green dots. (c) Schematic repre-

sentation of different graphs (chains, cycles, and path-redundant architectures) having a comparable longest path. The reference nodes for which both the

communicability (Cij) and the topological similarity (T ij) where calculated are highlighted in yellow. (d) Upper panels: comparison of Cij and T ij of the

three different graphs having a comparable longest path. Line colors correspond to those in the schematic representation in (c) (Light blue lines: chains;

red lines: cycles; dark green lines: two redundant paths; light green lines: three redundant paths; orange lines: four redundant paths). Lower panels: differ-

ence between path-redundant motifs and chains of the same length on the resulting Cij and T ij. All results were obtained for constant links’ weights and

global coupling w ¼ g ¼ 1.

047409-4 Bettinardi et al. Chaos 27, 047409 (2017)

In chains, both Cij and T ij decay with distance, blue lines

in Figure 1(d). On the other hand, the effect of the direct link

is well illustrated in the case of cyclic architectures (bottom-

right panels in Figure 1, red lines). The presence of a direct

link importantly enhances Cij and T ij and poses a lower bound

for them while the contribution due to the indirect path

decreases with its length.

The effect of path redundancy is best understood

when analyzing the difference between path-redundant motifs

and chains of the same length. The two lower panels in Figure

1(d) illustrate three examples of these differences, namely, the

cases with two (dark green lines), three (light green lines), and

four (orange lines) redundant paths. From this analysis, it

becomes evident that increasing the number of alternative

paths does enhance both the total influence and the topologi-

cal similarity between the end nodes and that the magnitude

of this increase decays with the length of the paths.

The results of this section show how the weight of the

links, the path length, and the presence of alternative routes

shape the manner in which the influences of nodes unfold

through the graph. These in turn define the most likely pat-

tern of interactions that a network is expected to sustain due

to its path structure.

IV. ANATOMICAL ROOTS TO FUNCTIONALCONNECTIVITY

In Section III, we have analyzed how very simple topo-

logical features shape the interaction between nodes. Real

networks, however, are made of intertwined assemblies of

those features, forming intricate architectures. As an exam-

ple of a real complex network, we now study how the ana-

tomical connectivity sculpts the complex pattern of

correlations observed in spontaneous brain activity. To this

aim, we calculate the topological similarity matrix, T SC, out

of the group-average structural connectivity (SC) and con-

sider it as the zero-order approximation of the correlations

due to the anatomical path structure. We then compare T SC

to the empirically observed resting-state functional connec-

tivity (FC). The SC matrix is a representation of the brain’s

wiring diagram, where axonal pathways are reconstructed

through diffusion tensor imaging (DTI) and tractography.

Despite the reproducibility of current tractography methods,

their accuracy to detect crossing fibers and long

interhemispheric axons is known to be limited.44–46 For these

reasons, we perform the analyses twice: for a single hemi-

sphere, Fig. 3(a), and for the whole brain, Fig. 3(b). The

details of the experimental procedures are described in Sec.

VI.

Adding a uniform coupling strength factor g to the

weights of the links allows us to scale how deep do influen-

ces propagate into the network (see Sec. II). When g is weak,

perturbations quickly decay producing local correlations

only around the node’s neighborhood. As g grows perturba-

tions propagate deeper into the network, giving rise to

stronger correlations over more distant nodes. Figure 2 illus-

trates three instances of T SC obtained for g¼ 0.35, 1 and 2

using the group-average SC matrix of the left hemisphere.

For the SCs corresponding to the whole brain and to the

left hemisphere, we have scanned through a broad range of

global coupling values g to find those returning the T SC that

best approximates the empirical FC, Fig. 3. We quantify

the approximation between T SC and the empirical FC by

means of the mean absolute error (MAE), an outlier-robust

alternative of the mean squared error, a classic statistic for

the goodness of an estimator. Compared to the similarity

between the raw SCs and the empirical FCs (dotted lines

in panels G and H of Fig. 3), T SC approximates the empirical

FC much better, passing from EðSC;FCÞ¼0:42 to EðT ;FCÞ¼0:15 for the calculations in the left-hemisphere. For the

whole brain network, it improves from EðSC;FCÞ¼0:44 to

EðT ;FCÞ¼0:22. The reason behind this improvement is

that, as mentioned earlier, the correlation between two

nodes is determined not only by the existence of a direct link

between them but also by all the common inputs they

receive, which depends on all the possible routes through

which information can travel. By definition, T SC accounts

for the effect of collateral influences traveling also along

indirect paths, while SC only represents direct links between

nodes. Moreover, the link weights associated with the SC

tracts do not necessarily “predict” the magnitude of the

empirical correlation between a pair of brain regions (see

supplementary material, Fig. S4).

A. Adding local dynamics

We now introduce local dynamics to the brain regions

and simulate their activity by means of the Hopf normal

FIG. 2. Effect of the global coupling. Effect of varying the global coupling g onto the resulting topological similarity matrix, T . The three matrices have been

obtained from the same empirical structural connectivity matrix (left hemisphere), for g¼ 0.35, 1 and 2.

047409-5 Bettinardi et al. Chaos 27, 047409 (2017)

model.47,48 We then compare the numerically simulated

functional connectivity of the whole network with the empir-

ical one. The Hopf model relies on the choice of a single

parameter, a, controlling the dynamical working-point of

each node, see Sec. VI for details. We set a¼ 0, meaning

that all nodes lie at the edge of the bifurcation, a regime that

has been demonstrated to give a good approximation of the

empirical resting-state FC and captures the properties of its

temporal fluctuations as well.49 As done for the topological

similarity, we scanned for a range of global coupling values

to find the g for which the simulated FC best approximated

the empirical FC, panels (A) and (B) in Fig. 4. The introduc-

tion of local dynamics gives rise to a correlation structure

that is, as expected, closer to the empirical FC than the

approximation using T : indeed, the mean absolute error falls

to E¼ 0.11 in the case of the left hemisphere and to 0.13 for

the whole brain network. The magnitude of the improvement

is however small. Together with the strong relationship

between the best-fitting simulated FC and T (see Figure 5),

it suggests that most of the similarities between the empirical

and the simulated FCs appear to be substantially shaped by

the underlying network architecture.

These results demonstrate that knowledge of the topol-

ogy of whole-network input patterns of different brain

regions, sustained by direct and indirect routes of multiple

interweaved axonal bundles, can be used to approximate the

time-average correlation structure observed from spontaneous

BOLD fluctuations beyond the information about direct ana-

tomical connections stored in the SC matrices.

V. SUMMARY AND DISCUSSION

In the present paper, we have studied the contribution

of topology in shaping the emergent pattern of interactions

a network tends to generate. In particular, our goal was to

identify the fundamental features behind that contribution.

We have shown how three primitive motifs (the strength of

the links, the length of the path, and the number of redun-

dant paths) regulate the expected interaction between two

nodes, as they alter the overall path structure of the net-

work. A strong direct link is usually a reliable indicator of

the magnitude of their interaction. However, the presence

of common inputs or redundant paths between them may

enhance their interaction beyond the baseline determined

by the direct link. When there is no direct link between the

nodes, common inputs and redundant paths can still trigger

strong correlations between them, although this tends to

decrease with the length of the paths. These results show

that what truly matters to describe topology’s influence on

the network dynamics is the global path structure of the net-

work and the weights assigned to each alternative path,

since it is unlikely to assume that information would propa-

gate only through shortest paths. Many efforts have been

FIG. 3. Contribution of whole-network common inputs to the brain’s spontaneous correlation structure. The figure shows results obtained separately for only

one hemisphere (Left panels) and for the whole brain (Right panels). (a), (b) structural connectivity matrices (SC). (c), (d) Scatterplots depicting the relation-

ship between SC and the empirical functional connectivity (FC). (e), (f) Empirical functional connectivity matrices (FC). (g), (h) Mean absolute error (MAE)

between the empirical FC and the topological similarity T computed for different values of the global coupling parameter, g. The dotted lines correspond to

the mean absolute error between the raw SC matrix and the empirical FC. (i), (l) Scatterplots of the empirical FC and the best-fitting topological similarity.

(m), (n) Best-fitting Topological similarity matrices. We included also the Pearson’s correlation values r corresponding to the best-fitting matrices obtained

optimizing the mean absolute error.

047409-6 Bettinardi et al. Chaos 27, 047409 (2017)

FIG. 4. Numerical simulations. The figure illustrates the results obtained from numerical simulations using the Hopf normal model. (a), (b) As for data

described in Section IV, the quality of the approximation was measured using the mean absolute error (MAE). MAE between empirical structural (SC) and

functional connectivity (FC) matrices is indicated by the dotted lines (�0:42 in both hemispheres). (c), (d) scatter plots of the empirical functional connectivity

(FC) versus the simulated one obtained at the best-fitting global coupling. (e), (f) In both panels, the upper triangles store the empirical FC values, whereas the

lower triangles the corresponding ones obtained from simulations at the best fitting value of g.

FIG. 5. Relationship between topologi-

cal similarity and simulated FC. The

two panels show the strong relation-

ship between the topological similarity

T and the Hopf-simulated FC (all

nodes having a¼ 0), both separately

optimized for the global coupling fac-

tor g. MAE between T and the Hopf-

simulated FC obtained for one hemi-

sphere (Left panel) is MAE ¼ 0:12 and

the Pearson’s correlation between

them is r¼ 0.88; MAE between T and

the Hopf-simulated FC obtained for

the whole-brain (Right panel) is MAE

¼ 0:19 , and the Pearson’s correlation

between them is r¼ 0.86.

047409-7 Bettinardi et al. Chaos 27, 047409 (2017)

devoted in the past to identifying the most influential topo-

logical features for the synchronizability and the spreading

dynamics on networks. Both phenomena have been reported

to depend on the heterogeneity of node degrees,50,51 the clus-

tering coefficient,52 the degree correlations,53,54 the between-

ness centrality,55 the k-coredness,56 and the community

organization.57,58 Our observations indicate that the mecha-

nism by which specific features, e.g., hubs, clustering or

communities, affect the collective dynamics is through their

power to alter the global path structure of the network.

We have developed a graph measure, the topological simi-larity T , which estimates the expected cross-correlation

between nodes based on the similarity of the estimated

“influences” two nodes receive from the whole network. If two

nodes receive the same sets of inputs, then they will tend to be

strongly correlated. In graph analysis, the similarity of nodes is

typically characterized by the matching index, which is calcu-

lated as the number of common neighbors shared by two

nodes. However, the matching index only accounts for the

direct links and ignores the indirect paths. The topological sim-

ilarity is a generalization of the matching index which accounts

for the weighted influence that every node receives from all

the others along all possible paths, including recurrent and

redundant ones. Indeed, it has recently been shown that the

presence of ensembles of alternative paths seems to increase

the overall network resiliency.39

When introducing T , we have stated that the topological

similarity can be regarded as the expected cross-correlation

matrix emerging due to the underlying network structure. In

fact, it was recently shown that considering communicability

as the propagator kernel for a linear system of noise diffu-

sion, the time-average cross-correlation matrix R of the sys-

tem could be analytically approximated.43 We have found

that both approaches, the one starting from a dynamical

system describing the propagation of perturbations and the

topological one presented here, are indeed equivalent. The

equivalence holds when all nodes are identical, i.e., have the

same relaxation time-constant, and when the Gaussian noise

fed into each node has the same variance and amplitude (see

supplementary material).

A. Structure-function relation in the human brain

As a practical example, we have explored the contribu-

tion of the anatomical architecture to the spontaneous corre-

lation structure measured with functional imaging. To this

aim, we have calculated the topological similarity, T SC, out

of the tractography-based structural connectome (SC) and

have compared it to empirically obtained functional connec-

tivity (FC) from resting-state fMRI. See also Fig. S5 (supple-

mentary material) for a comparison between the different

abilities of the communicability and the topological similar-

ity in numerically approximating the empirical FC. The

result shows that T SC partly captures the empirical FC with-

out fully reproducing it. This difference is to be expected,

as the collective network dynamics depend do not only on

the underlying anatomical connectivity but also on the

state-dependent dynamical regimes of brain regions (e.g.,

awake state, sleep, or anesthesia) and the manner in which

information is passed from one to another. By definition,

topological similarity does not account for those dynamical

factors; it captures the propensity of nodes to correlate or

synchronize, thanks to the topology of the network they are

embedded into. This tendency will then be modulated by the

particular local dynamics and the coupling function of the

system at hand.

Recently, a similar formalism as the one we used here

has been applied but intended to find an optimal fit between

the structural and the functional connectivities.59,60 The

underlying assumption in these works is that the empirical

FC matrix F can be directly estimated from the weighted

adjacency matrix A of the SC, such that there exists a func-

tion f which transforms F ¼ f ðAÞ. Given f to be the power

seriesP1

l¼0 blAl, the goal is thus to estimate the coefficients

fblg leading to a best fit between F and the empirical FC. It

shall be noted, however, that the optimal coefficients found

in such cases are difficult to interpret from a physical per-

spective. Because F also depends on the local dynamics and

the coupling function (let’s denote them as M and H respec-

tively), the actual estimation problem should be defined as

F ¼ f ðA;M;HÞ. By assuming that F depends only on A such

that F ¼ f ðAÞ, the optimal coefficients intrinsically carry

information about the other hidden variables M and H which

were ignored from the optimization problem. It shall be

emphasized that the rationale behind T is not that of opti-

mizing F as a function of A but instead it is that of providing

a theoretical explanation for the mechanism through which

the structure of a network shapes the propensity of nodes to

correlate with each other.

Additionally, we have simulated the resting brain

activity by modeling the local node dynamics as nonlinear

units with external noise. The FC resulting from the simula-

tions improves the accuracy to approximate the empirical

FC beyond the structural contribution described by T SC.However, it does not fully reproduce the empirical observa-

tion. Our results are in accordance with previous models of

resting-state brain activity whose ability to replicate the

empirical FC has been systematically reviewed in Refs. 10

and 27. The whole-brain network models proposed so far

could only capture the empirical FC to a limited degree,

with the closest models exhibiting a correlation of r � 0:6between simulated and empirical FC. These results indicate

that many of the factors determining the generation of spon-

taneous brain fluctuations still elude our ability to capture

them. Whether these limitations are due to the local model

selection or other reason, e.g., the precision of tractography

to identify fiber tracts, remains a question for further investi-

gation, although the existence of different state-dependent

correlation patterns (as seen in rest, sleep, or anesthesia)

sustained by the same underlying network clearly demon-

strate the crucial role of dynamical factors in determining

the emergent collective activity. Accordingly, interpretation

of the topological similarity here introduced as a model to

reproduce the empirical FC (either at rest, sleep, or under

anesthesia) shall be avoided, since its role is that of exposing

the expected patterns of correlation due to the anatomical

structure alone, patterns which are known to be altered by

the global brain state.

047409-8 Bettinardi et al. Chaos 27, 047409 (2017)

B. Limitations

The results presented in this paper come with some limi-

tations which shall be emphasized. (i) Topological similarity

can be regarded as a topological estimation of the expected

time-averaged cross-correlation. Thus, it is by definition

blind to transient temporal fluctuations in the correlation

patterns which may emerge in the network. (ii) We have

here based the calculation of T on the communicability mea-

sure41 which assumes an arbitrary decay of the influence

with the length of the path. Although the precise decay rate

of the signals may differ across real systems, our choice of

the communicability guarantees that the accumulation of

influence, Eq. (2), converges for all adjacency matrices A. In

principle, the concept of topological similarity is indepen-

dent of the definition of communicability; any other decay

of the influence along the path can be used to estimate T ,

as long as the chosen set of coefficients fblg ensure the con-

vergence of the power-series for the adjacency matrix and

monotonically decays with l. (iii) For the analysis on the

brain’s connectivity and spontaneous correlations, we con-

sidered both hemispheres as if they were independent. The

motivation was to avoid biases due to the unreliability of

tractography to identify inter-hemispheric fibers. Still, our

comparisons are biased to some extent because the SC-based

T and simulations consider both hemispheres as independent

while the empirical resting-state measurements reflect the

activity of the brain regions, which are certainly embedded

on the whole network.

VI. MATERIALS AND METHODS

Twenty-one healthy volunteers (mean age 21.56 years;

standard deviation 1.84 years; all males; all right handed)

participated in five (5) resting-state and two (2) DTI scan-

ning sessions and signed an informed consent. The study

was conducted in the School of Psychology, Birmingham

and was approved by the University of Birmingham Ethics

Committee.

A. Data acquisition

Scanning sessions were conducted at the Birmingham

University Imaging Centre using a 3 T Philips Achieva MRI

scanner with a 32-channel head coil. T1-weighted anatomical

data (175 slices; 1� 1� 1 mm3 resolution) were collected

during the first scanning session and DTI data were collected

in two sessions (23.3 6 2.5 days apart). The DTI acquisition

consisted of 60 isotropically distributed diffusion weighted

directions (b¼ 1500 smm�2; TR¼ 9.5 s; TE¼ 78 ms; 75 sli-

ces; 2� 2� 2 mm3 resolution; SENSE) plus a single volume

without diffusion weighting (b¼ 0 smm�2, denoted as b0).

The DTI sequence was repeated twice during each session,

once following the Anterior-to-Posterior phase-encoding

direction and once the Posterior-to-Anterior direction, to

correct for susceptibility-induced geometric distortions.61

Resting-state data were collected in five sessions (the first and

the last collected in the same scanning session as the DTI

data) using whole brain echo-planar imaging (EPI) (180

volumes; TR¼ 2 s; TE¼ 35 ms; 32 slices; 2.5� 2.5� 4 mm3

resolution). Participants were instructed to have their eyes

open and maintain fixation to a white dot presented at the cen-

tre of the screen.

B. Whole-brain DTI tractography

We processed the DTI data in FSL version 5.0.8

(FMRIB Software Library, http://fsl.fmrib.ox.ac.uk/fsl/

fslwiki/). We first corrected the data for susceptibility distor-

tions, eddy currents, and motion artifacts62 and rotated the

gradient directions (bvecs) to correct them for motion rota-

tion.44,63,64 We then generated a distribution model in each

voxel using FSL BedpostX65 with default parameters.

We parcellated the brain into 116 areas using the

Automated Anatomical Labeling (AAL) atlas.66 We fol-

lowed a 4-step registration procedure to align the AAL

atlas from Montreal Neurological Institute (MNI) template

to native space: (a) align the non-weighted diffusion volume

(b0) of each session to their midspace and create a midspace-

template (rigid-body),67,68 (b) align the midspace-template to

the anatomical (T1) scan (rigid-body), (c) align the T1 to the

MNI template of FSL (non-linear), and (d) invert and combine

all the transformation matrices of the previous steps to obtain

the MNI-to-native registration. The final matrix was applied to

the AAL atlas (nearest-neighbour interpolation was used in

order to preserve discrete labeling values). The results of each

step were visually inspected to ensure that the alignment was

successful.

We simulated tracts (i.e., probabilistic streamlines) start-

ing from each AAL area and reaching any other AAL area

using the Probabilistic Tracking algorithm (ProbtrackX).69

The parameters we used in ProbtrackX are: 5000 samples

per voxel, 2000 steps per sample until conversion, 0.5 mm

step length, 0.2 curvature threshold, 0.01 volume fraction

threshold and loopcheck enabled to prevent streamlines from

forming loops. We normalised the tracts by the size of the

seed area and thresholded the normalised tracts at 1% of the

maximum value (i.e., setting them to zero). We subsequently

computed the undirected structural connectivity matrix by

averaging the normalised tracts from area i to area j and

from area j to area i, as directionality of the reconstructed

fiber tracts cannot be inferred from DTI.

C. Population-average structural connectome

To estimate the population average structural connectiv-

ity (SC), we pooled the 42 SC matrices together (2 per sub-

ject) and considered both the complete parcellation having

cortical, subcortical, and cerebellar regions of interest (ROIs),

and the reduced one (90 brain areas—45 per hemisphere—,

which excluded the 26 regions of the cerebellum and the ver-

mis). The 42 SC matrices for both parcellation contained a

variable number L of undirected links ranging from L¼ 895

for the sparsest case (density q ¼ 0:22) to L¼ 1279 for the

densest (q ¼ 0:32). We noticed that the simple average of the

matrices into a single SC matrix by averaging the 42 values,

each link taken along the pool leads to an average connectome

with strongly biased network properties. For example, this

plain average SC matrix contained L¼ 1967 links, which are

almost twice the number of links as in the individual matrices.

047409-9 Bettinardi et al. Chaos 27, 047409 (2017)

In order to avoid this problem, we have devised a method

which automatically removes the outlier links before perform-

ing the average. For each link (i, j), we have initially a set of

42 weights fwsijg where s ¼ 1; 2;…; 42. The method searches

for outlier weights (data-points falling out of 1.5 times the

inter-quartile range) and removes them from the data pool.

The search is iteratively repeated until no further outliers are

detected and then the population-average SC weight for the

link (i, j) is calculated as the average weight of the surviving

values. In practice, the method converges very rapidly, and it

rarely performs more than 2 iterations per link. This method

allows us to clean the data without having to set an arbitrary

hard threshold70 for the minimally accepted prevalence of the

link. Full details of the method are currently in preparation

and will be presented somewhere else. The resulting

population-average SC matrix out of our iterative pruning

method contains L¼ 1189 links (q ¼ 0:30), which lies within

the range of connectivity for the individual 42 matrices.

D. Resting-state time-courses and functionalconnectome

We pre-processed the EPI resting-state data in FSL ver-

sion 5.0.8 (FMRIB Software Library, http://fsl.fmrib.ox.ac.uk/

fsl/fslwiki/) using MELODIC (Multivariate Exploratory Linear

Optimized Decomposition into Independent Components). We

corrected the data for motion and slice scan timing, removed

the non-brain tissue, applied 5 mm FWHM spatial smoothing

and removed spike motion artifacts using WaveletDespike.71

We subsequently applied high-pass temporal filtering and

then extracted the average timecourse from each AAL area. To

estimate the population-average functional connectivity (FC)

matrix, we concatenated the 105 sequences of resting-state sig-

nals (21 subjects, 5 sessions per subject) into a single long mul-

tivariate time-series and computed the Pearson correlation for

every pair of signals. The opposite procedure, to compute an

FC matrix per session and averaging over the 105 FC matrices,

leads to almost identical results.

E. Hopf normal model

Within this model, the temporal evolution of the activity

z of node j is given in the complex domain as

dzj

dt¼ aj þ ixj � jz2j� �

þ rgj tð Þ; (5)

zj ¼ qjeihj ¼ xj þ iyj; (6)

where x is the node’s intrinsic frequency of oscillation, a is

the local bifurcation parameter (local because the model

allows the possibility to assign a different value of a for each

node in the network), and g is the additive Gaussian noise

with standard deviation r. This system has a supercritical

bifurcation at a¼ 0. If aj < 0, then the local dynamic has a

stable fixed point at zj¼ 0, while for aj > 0, the nodes follow

a stable limit-cycle oscillation of frequency f ¼ x=2p.

Whole-brain dynamics are described by the following cou-

pled equations:

dxj

dt¼ ½aj � x2

j � y2j �xj � xjyj þ g

XN

i¼1

Cij xi � xjð Þ þ rgxj tð Þ;

(7)

dyj

dt¼ ½aj � x2

j � y2j �yj þ xjxj þ g

XN

i¼1

Cij yi � yjð Þ þ rgyj tð Þ;

(8)

where Cij is the anatomical connectivity between nodes i and

j, g is the global coupling factor, and the standard deviation

of gaussian noise is r¼ 0.02. In this model, the simulated

activity corresponds to the BOLD signal of each node. The

intrinsic frequency of each node was estimated as the peak

frequency in the associated narrowband (i.e., 0.04–0.07 Hz

(Ref. 72)) of the empirical BOLD signals of each brain

region. We simulated, for each of the two hemispheres (45

ROIs each), 330 000 points using Euler’s method for integra-

tion (dt¼ 0.001). The connectivity between all the regions of

interest was defined using the empirical structural connectiv-

ity matrix (SC) and obtained time-series were then used to

compute the simulated correlation matrix (Simulated FC) by

Pearson cross-correlations of the resulting time series.

SUPPLEMENTARY MATERIAL

See supplementary materials for the formal demonstra-

tion of the equivalence between T and R when all network

nodes are assumed to be identical, and they all receive a

Gaussian white noise of the same intensity and variance,

together with five supplementary figures: the first supplemen-

tary figure illustrates the effect of direct single link’s weight

on both the communicability and the topological similarity

between the two nodes; the second supplementary figure

shows the quantitative effect of varying links’ weights in

chain and cyclic topologies; the third supplementary figure

depicts the behavior of four different measures of model

fitting, namely Euclidean distance, mean squared error,

mean absolute error, and Pearson’s correlation coefficient; the

fourth supplementary figure demonstrates the capacity of T to

capture also the influence of indirect paths on the correlation

observed between brain regions that, according to the empiri-

cal SC matrix, are not directly connected; the last supplemen-

tary figures show the difference between communicability

and topological similarity in numerically approximating the

empirical correlation structure.

ACKNOWLEDGMENTS

We would like to thank Rui Wang and Caroline di

Bernardi Luft for their help in collecting the data used in this

study. This work was supported by (R.G.B.) the FI-DGR

scholarship of the Catalan Government through the Agencia

de Gesti�o d’Ajuts Universitari i de Recerca, under Agreement

No. 2013FI-B1-00099, (G.Z.L.) the European Union’s

Horizon 2020 research and innovation programme under Grant

Agreement No. 720270 (HBP SGA1), (G.D.) the European

Research Council Advanced Grant: DYSTRUCTURE

(295129) and the Spanish Research Project No. PSI2013-

42091-P, (Z.K.) European Community’s Seventh Framework

047409-10 Bettinardi et al. Chaos 27, 047409 (2017)

Programme [FP7/2007-2013] under agreement PITN-GA-

2011-290011, (V.M.K.) European Community’s Seventh

Framework Programme [FP7/2007-2013] under Agreement

No. PITN-GA-2012-316746 and (M.L.K.) by the European

Research Council Consolidator Grant No. CAREGIVING

(615539).

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